A vector sensor can provide measurements of ocean acoustic fields in terms of the acoustic pressure and three-dimensional particle velocity, providing potentially highly-informative data for applications such as geoacoustic inversion. This paper applies nonlinear Bayesian inversion to vector sensor data to estimate seabed geoacoustic properties and uncertainties in South China Sea. Linear-frequency-modulated source transmissions, recorded as acoustic pressure and vertical particle velocity, are processed to estimate the vertical phase gradient of acoustic pressure at multiple frequencies as the inversion data. An advantage of this type of data is that it can be modeled without knowledge of the source spectrum, allowing inversion with an unknown source and a single sensor. Geoacoustic inversion of phase-gradient data is carried out and compared to inversion of the vertical acoustic impedance, another type of vector-sensor data, independent of the source spectrum, which has been considered previously. Model selection for the optimal number of seabed sediment layers is carried out using Bayesian information criterion, and parameter estimates, uncertainties, and correlations are calculated using delayed-rejection adaptive Metropolis-Hastings sampling. Results indicate a three-layer seabed model (including the semi-infinite basement), with properties in agreement with independent measurements including a high-resolution seismic profile and surficial sediment type from a core.